Respuesta :

LammettHash

C. For each term in this expression, you can just apply the power rule.

[tex]y = 3x^2 - \sqrt x + \dfrac2x = 3x^2 - x^{\frac12} + 2x^{-1}[/tex]

[tex]\implies \dfrac{dy}{dx} = 2\cdot3x^{2-1} - \dfrac12 x^{\frac12-1} + (-1)\cdot2x^{-1-1}[/tex]

[tex]\implies \dfrac{dy}{dx} = 6x - \dfrac12 x^{-\frac12} - 2x^{-2}[/tex]

[tex]\implies \dfrac{dy}{dx} = 6x - \dfrac1{2\sqrt x} - \dfrac2{x^2}[/tex]

Every other choice involves composite functions that do require the chain rule to differentiate.

Option C  : [tex]y = 3x^2 - \sqrt{x} + \dfrac{2}{x}\\[/tex]  can be differentiated without using chain rule.

The differentiation of  [tex]y = 3x^2 - \sqrt{x} + \dfrac{2}{x}\\[/tex]   without using chain rule can be done as follows:

[tex]\begin{aligned}\dfrac{dy}{dx} &= \dfrac{d}{dx}(3x^2 - \sqrt{x} + \dfrac{2}{x})\\\\&= \dfrac{d}{dx}(3x^2 - x^{\frac{1}{2}} + 2x^{-1})\\&= 6x - (1/2)x^{-\frac{3}{2}} -2x^{-2}\\\end{aligned}[/tex]

The rest of the functions are composite, which means, that after differentiating them with some other base, you need to change the base and differentiate again, thus applying chain rule.

Thus, the function [tex]y = 3x^2 - \sqrt{x} + \dfrac{2}{x}\\[/tex]   can be differentiated without using chain rule.

Thus, option C:   [tex]y = 3x^2 - \sqrt{x} + \dfrac{2}{x}\\[/tex]   is correct option.

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https://brainly.com/question/5543508

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